In the study of temporal graphs, only paths respecting the flow of time are relevant. In this context, many concepts of walks disjointness were proposed over the years, and the validity of Menger's Theorem, as well as the complexity of related problems, has been investigated. In this paper, we introduce and investigate a type of disjointness that is only time dependent. Two walks are said to be snapshot disjoint if they are not active in a same snapshot (also called timestep). The related paths and cut problems are then defined and proved to be W[1]-hard and XP-time solvable when parameterized by the size of the solution. Additionally, in the light of the definition of Mengerian graphs given by Kempe, Kleinberg and Kumar in their seminal paper (STOC'2000), we define a Mengerian graph for time as a graph $G$ that cannot form an example where Menger's Theorem does not hold in the context of snapshot disjointness. We then give a characterization in terms of forbidden structures and provide a polynomial-time recognition algorithm. Finally, we also prove that, given a temporal graph $(G,\lambda)$ and a pair of vertices $s,z\in V(G)$, deciding whether at most $h$ multiedges can separate $s$ from $z$ is NP-complete.

翻译：在时间图的研究中，只有遵循时间流动的路径才具有相关性。在此背景下，多年来提出了多种关于路径不相交性的概念，并探讨了门格定理的有效性及相关问题的复杂性。本文引入并研究了一种仅依赖于时间的不相交性类型。若两条路径不在同一快照（也称为时间步）中活跃，则称它们为快照不相交。随后定义了相关的路径与割问题，并证明当以解的大小为参数时，这些问题为W[1]-难且属于XP时间可解。此外，根据Kempe、Kleinberg和Kumar在其开创性论文（STOC'2000）中给出的门格图定义，我们定义了时间上的门格图，即图$G$不能构成在快照不相交背景下门格定理不成立的实例。进而，我们给出了基于禁止结构的刻画，并提供了一种多项式时间识别算法。最后，我们还证明，给定时间图$(G,\lambda)$和一对顶点$s,z\in V(G)$，判定是否最多$h$条多边能分离$s$与$z$是NP完全的。